Geodesic
Arc-transitive regular cyclic covers of the complete bipartite graph $\mathsf{K}_{p,p}$
Journal of Algebraic Combinatorics, Tome 42 (2015) no. 2, pp. 619-633.

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Characterizing regular covers of edge-transitive or arc-transitive graphs is currently a hot topic in algebraic graph theory. In this paper, we will classify arc-transitive regular cyclic covers of the complete bipartite graph $\mathsf{K}_{p,p}$ for each odd prime $p$. The classification consists of four infinite families of graphs. In particular, such covers exist for each odd prime $p$. The regular elementary abelian covers of $\mathsf{K}_{p,p}$ are considered in a sequel.
Classification : 05C25, 05C70, 20B15, 20B30
Keywords: arc-transitive graph, regular cover, complete bipartite graph, Cayley graph 
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     title = {Arc-transitive regular cyclic covers of the complete bipartite graph $\mathsf{K}_{p,p}$},
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     pages = {619--633},
     publisher = {mathdoc},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2015__42_2_a1/}
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Pan, Jiangmin; Huang, Zhaohong; Liu, Zhe. Arc-transitive regular cyclic covers of the complete bipartite graph $\mathsf{K}_{p,p}$. Journal of Algebraic Combinatorics, Tome 42 (2015) no. 2, pp. 619-633. http://geodesic.mathdoc.fr/item/JAC_2015__42_2_a1/